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34 On the Rough Number Computation and the Ada Language 445

Table 34.1 Ada ﬂoating-point types

Precision

Type Size (bize) (digits)

f–ﬂoat 32 6

d–ﬂoat 64 9

g–ﬂoat 64 15

h–ﬂoat 128 33

In addition, most Ada compilers provide the types long ﬂoat and long long ﬂoat

(used in package standard) and f

ﬂoat, d ﬂoat, g ﬂoat,andh ﬂoat (used in package

system) [8, 14]. The size and the precision of each of the Ada ﬂoating-point types

are given in Table 34.1.

The goal of computation is accuracy. Higher accuracy will provide more reliabil-

ity in the real-time environment. Sometimes, a single precision or a double precision

of ﬂoating-point numbers in FORTRAN 77 [9] is not enough for solving some crit-

ical problems. In the Ada language, one may use the ﬂoating point number type:

long

long ﬂoat (h ﬂoat) by declaring “digit 33” to use 128 bits for ﬂoating point

numbers provided by Vax Ada [14] for example to achieve a precision of 33 deci-

mal digits accuracy; the range of the exponent is about from 10

134

to 10

C134

448 to C448 of base 2 [1,2,4–6, 8, 14] for the range.

The author employed this special accuracy feature in the computation of the hy-

pergeometric distribution function [16]. Similarly, for ﬁxed-point types,themodel

numbers are in the form:

sign mantissa  small: (34.10)

The sign is either C1 or 1; mantissa is a positive integer; small is a certain positive

real number.Themodel numbers are deﬁned by a ﬁxed-point constraint, the number

small is chosen as the largest power of two that is not greater than the delta of a ﬁxed

accuracy deﬁnition. The Ada language permits the user to determine or declare a

possible range and an error bound which is called delta for computational needs.

Examples are the follows:

Overhead has a delta of 0.01 and

Overhead has a range from  1:0E5 to 1:0E5: (34.11)

These indicate small is 0.0078125 which is 2

7

, and model numbers are in the

following interval

Œ12; 800; 000  small; C12; 800; 000  small: (34.12)

The predetermined range provides a reliable programming environment. The user

assigned error bound delta guarantees an accurate computation. These ﬂoating-

point number and ﬁxed-point number types not only provide good features for

446 T. Wu

real-time critical computations, but also give an extra reliability and accuracy for

general-purpose numerical computations. This result can be provided by interval

computation [11, 12] too. But using the primitive Ada programming method can

eliminate the more complicated implementation of interval computation.

34.7 Conclusions

In this chapter, we have reviewed some basic algebraic structures such as abelian

group, ring,andﬁeld. We deﬁned sets of rough numbers, dyadic numbers, ﬁnite

dyadic numbers, limited dyadic numbers,andmodel numbers; from these deﬁni-

tions, we have learned that the difﬁculty of numerical computation is one must

actually work with two distinct number systems. Solving any numerical compu-

tation problem consists of the following three parts: (1) the problem is given in the

real number system, (2) the computation is done in the model number system for the

given machine, and (3) the results must be converted from model numbers into real

numbers. Real numbers and model numbers have two different algebraic structures,

and they are not isomorphic. In general, starting from the problem to computation

on a machine can create some errors and going from the machine computation to

reporting results in real numbers can induce errors too. This chapter has reported

that a computer system is not capable of computing real numbers accurately within

its constraints from a theoretical viewpoint.

Computation over the set of real numbers requires performing a ﬁeld computa-

tion. However, when a real number within the given range is stored or read into a

computer system, it is converted into a dyadic number. Computation over the set of

limited dyadic numbers is dyadic number computation.

Let f be a mapping that takes each real number into its dyadic number repre-

sentation. If a and b are two real numbers within a given range and they map into

their dyadic representations f.a/and f.b/, respectively, then in general, we should

have f.a C b/ ¤ f.a/ ˚ f.b/. The addition, C, on the right-hand side is the ad-

dition of real numbers. The mapping f does not preserve the algebraic structure,

so the set of real numbers and the set of dyadic numbers are not isomorphic.To

avoid overloading and possible confusion, we will introduce a new addition ‘˚’for

the dyadic number addition. To adjust the inequality, we will add in an error term,

Err. Therefore, we have f.a C b/ D f.a/ ˚ f.b/C Err. To do this, we use the

special features of the Ada programming language either to declare variables with

a speciﬁc value of “small”forﬁxed-point numbers or to designate a value of delta

for ﬂoat-point numbers. In this way, the error of the computation result is controlled

within the acceptable level.

34 On the Rough Number Computation and the Ada Language 447

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14. Vax Ada language reference manual (1985) Digital Equipment Corporation, Maynard, MA

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Englewood Cliffs, NJ

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Author Index

Abbas, A.E., 77

Abbot, J., 422

Abdul Azeez, M.F., 121

Abela, R.L., 422

Aberth, O., 444–445

Ablow, C.M., 320

Addison, P.S., 280

Adiletta, G., 266

Aggarwal, H.R., 320

Aguado, M., 234

Ahmad, S., 332

Ajayan, P.M., 96, 101–102

Akay, A., 95

Al-Bender, F., 97

Albert, R., 203

Alefeld, G., 444

Ali, M.S., 135

Alleyne, D., 341, 352

Ambr´osio, J., 358

Anagnostopoulos, A.N., 106

Andrews, R., 99

Antoni, J., 407

Apsel, R.J., 320, 332

Arakere, N., 105

Arimoto, S., 157

Arnold, B.J., 422–423

Arnold, R.N., 323, 334

Ashley, H., 62

Atluri, S.N., 358

Averina, V., 135

Awojobi, A.O., 323–324, 331, 333–334

Bag, B.C., 224

Bagley, R.L., 3

Baier, G., 31

Bainov, D., 204, 210

Bainton, K., 348

Bakis, C.E., 96

Banavar, J.R., 217

Banerjee, P.K., 320–321

Barab´ai, A.L., 203

Barkan, D.D., 320, 331, 333

Barnes, J.G.P., 438, 441, 444–445

Bartuccelli, M., 106

Bautin, A.N., 157

Bechtel, S.E., 293–295

Beeston, H.E., 334

Benner, H., 169

Bently, D.E., 121

Benzi, R., 251

Beran, J., 23

Beredugo, J.O., 333

Bergman, L.A., 372, 377

Bernard, T., 169

Beskos, D.E., 331

Bethell, P.J., 422–423

Bettess, P., 331

Bezrukov, S.M., 224

Bielak, J., 358

Bindemann, A.C., 358

Bishop, S., 177–178, 184

Black, C., 280

Black, H.F., 121

Bonilla, B., 4–7

Bouwman, V., 385

Boyd, R.W., 234

Brackbill, C.R., 95

Braun, H.A., 169–170

Buckingham, E., 348

Bueler, E., 135

Butcher, E.A., 135

Butterﬁeld, R., 331

Bycroft, G.N., 323–324, 334

449

450 Author Index

Caldas, I.L., 106

Campbell, S.A., 159

Cao, D.-Q., 263–272

Cao, L., 224, 234, 236

Carlson, C.D., 293–295

Carmona, R., 279

Carnahan, B., 140

Carrol, T.L., 13

Casadememunt, J., 234

Casademunt, J., 234

Castellani, A., 324

Castro, F., 223–224

Cawley, P., 341, 352

Cawley, R., 32

Chae, Y.S., 334

Chan, C.K., 334

Chandler, D.E., 321

Chandrasekaran, H., 77

Chao, C.C., 320

Chapel, F., 332

Chaplin, C.R., 291

Chavz, A., 124

Chen, C.H., 136

Chen, C.W., 409

Chen, D.M., 384

Chen, G.R., 158, 203

Chen, S.B., 234

Chen, W.H., 136

Chen, X.F., 224

Chen, Y.-H., 13, 33

Cheng, D.M., 385

Chesse, P., 397

Childs, D.W., 121

Chinvorarat, S., 417

Choi, S.K., 121

Choi, Y.S., 121

Chongbin, Z., 331

Chow, S.N., 385

Chow, Y.K., 329

Christiansen, P.L., 106

Chu, F., 275

Chu, T., 409–435

Chua, D.H.C., 96

Chua, L.O., 31, 204

Chueshov, I.D., 385

Chuhan, Z., 331

Chui, C.K., 278

Chung, K., 178

Cipra, R.J., 295

Cipric, G., 177

Claro, J.C.P., 358

Clemmet, J.F., 324, 333

Cohen, I., 31

Coleman, J.N., 99

Collinger, J.C., 95

Costello, G.A., 285, 287

Costello, H., 358

Cotsaftis, M., 291, 300, 313

Cox, D.E., 410

Crede, C.E., 50

Crowe, C.T., 430

Cunny, R.W., 334

Curran, S.A., 99

Curry, J.H., 37

Cvitanovic, P., 37

Dai, G.-P., 251

Dalby, A.R., 203

Danca, M.F., 191

Daniere, J., 407

Das, S., 3, 5

Davies, T.G., 320

Davies, T.J., 286, 289–290

Davison, D.E., 136

Dawance, G., 334

Day, S.M., 331

Day, W.B., 121

de Jongh, J., 384

de Kraker, A., 384

de Souza, S.L.T., 106

Deck, J.F., 358

Deng, Y., 251

Deshmukh, V., 135

Devulder, C., 264

Dhamala, M., 158

Dickey, E.C., 99

Diez-Martines, O., 170

Ding, J., 263–264

Ding, M., 158

Ding, Q., 264

Dion, J.M., 136

Ditto, W.L., 32

Dixit, S.N., 233

Dobry, R., 330–331

Dominguez, J., 331

Don, J., 409–435

Dong, S., 342, 344–346

Dowell, E.H., 62

Dragana, T., 158

Driver, D., 183

Drnevich, V.P., 334

Duan, L., 158

Duan, Y.W., 13

Dubow Sky, S., 358

Dugard, L., 136

Author Index 451

Dugundji, J., 62

Duns, C.S., 331

Durand, D.M., 217

Eastwood, W., 334

Eckman, J.P., 13

Ehrich, F.F., 121

Elata, D., 288

Elbeyli, O., 136, 138, 140, 144–146

Elger, D.F., 430

Elorduy, J., 320, 324

Emami-Naeini, A., 143

Eshkenazy, R., 288

Ettouney, M.M., 331

Fan, D., 157–166

Fan, S.-B., 251–259

Fang, T., 32

Faria, T., 159

Fedigan, S., 122

Feldmann, A., 23

Feng, N.S., 264

Feng, R., 169–174

Feng, T., 280

Feng, Z., 158

Ferri, A.A., 357–358

Fey, R.H.B., 384

Findlay, W., 438, 441–442

FitzHugh, R., 157

Flores, P., 358

Foias, C., 264, 385

Fox, R.F., 235–236

Frank, M., 234

Franklin, G.F., 143

Frazier, G.A., 331

Freitas, M., 177

Friedland, S., 347

Friswell, M.I., 384

Fry, Z.B., 332, 334

Fu, X.L., 203–215

Fuentes, M.A., 224–226

Fujimoto, S., 297

Fujisaka, H., 236

Fulinski, A., 223

Gallas, J.A.C., 37

Galvanetto, U., 177–178, 184

Gan, C., 177–178

Gao, G., 394

Garcia-Archilla, B., 264–265

Garg, N.K., 135

Garvey, S.D., 384

Gaul, L., 358

Gazetas, G., 331

Gazis, D., 342, 347

Ge, S., 287

Gee, D.J., 62

Gegg, B.C., 77–93

Gerlach, U.H., 374

Ghosh, P.K., 224

Gilbert, A.C., 23

Girard, J., 331, 333

Gladwell, G.M.L., 323

Gokcek, C., 191

Goldman, P., 121

Golinval, J.C., 275

G´omez, X., 291

Goncalves, P.B., 135

Gonchenko, S.V., 37

Gonchenko, V.S., 37

Gong, Y.F., 170

Goswami, G., 224

Goto, S., 203

Goychuk, I., 224

Grebogi, C., 32, 105, 109, 170, 177

Grebogi, E.C., 32

Grifﬁn, J.H., 358

Grootenhuis, G., 324–325, 330, 333–334

Grootenhuis, P., 323, 334

Grossmann, S., 236

Grune, L., 191

Gu, K., 136

Guan, Z.H., 136

Guckenheimer, J., 384

Gudyma, Y.V., 223

Guido, A.R., 266

Guillot, M., 334

Gunaratne, G.H., 37

Guo, S., 159

Guo, Y., 37–47

Gupta, D.C., 333

Gustavsson, I., 409

Guyan, R.J., 264

Hai, W.H., 13

Hakal, P.D., 191–192

Haken, H., 223

Hall, J.R. Jr., 330, 332–333

Hamidzadeh, H.R., 319–335

Han, Q.L., 136

452 Author Index

Han, R.P.S., 38

Han, X.P., 203–215, 342

Hanggi, P., 236

H¨anggi, P., 224

Hardin, B.O., 334

Harding, J.W., 323

Harris, C.M., 50

Hashemi, S.M., 297

Hassard, B.D., 159, 163, 165

Hay, T., 341

He, D.H., 13, 33, 217–219, 221

He, K., 106

He, L., 50

He, Q., 32

He, Y.N., 136, 386

Heller, L.W., 320

Henon, M., 37

Hernandez-Garcia, E., 234

Hernandez-Machado, A., 234, 240

Herstein, I.N., 438–439

Herzberger, J., 444

Hespanha, J.P., 136, 145–146

Hetet, J.F., 397

Hildebrand, F.B., 63

Hindmarsh, J.L., 171

Hiroshi, K., 158

Hisayo, M., 158

Hodgkin, A.L., 157

Hodgman, T.C., 203

Hoelscher, J., 421–435

Holmes, P., 384

Honaker, R.Q., 422–423

Hong, D.W., 295

Hong, L., 31–36, 121–131, 157–166, 217–221

Hope, R., 122

Horne, A.B., 203

Horsthemke, W., 223

Hou, Z.K., 135

Housner, G.W., 324

Hruska, S.L., 37

Hsiao, M.-H., 410

Hsieh, T.K., 330, 332

Hsu, C.S., 32

Hsu, S.T., 358

Hu, G., 223, 227

Hu, H.Y., 4, 162, 169

Hu, S.J., 170

Hu, W., 264

Hu, X.B., 136

Huang, H.C., 136

Huang, J.H., 96

Huang, J.-K., 409–410, 417

Huang, K., 178

Huang, L.H., 159, 162, 371

Huang, N.E., 251, 398, 401

Huang, W.H., 263–272

Huang, W.L., 13

Huang, Y., 95–103

Huang, Y.P., 289–290

Huang, Y.S., 159

Huber, M.T., 169–170

Huberman, B.A., 170

Hui, D., 96

Huxley, A.F., 157

Hwang, W., 279

Ikeda, K., 384

Iljitchov, V.A., 333

Inozu, B., 397

Inse, G., 158

Insperger, T., 140, 145

Israil, A.S.M., 332

Itoh, M., 204

Iwan, W.D., 97–98

Jackson, J.K., 23

Jacob, K.I., 293–295

Jalili, N., 96

Janbu, N., 330

Jauberteau, F., 385

Jeong, H., 203

Ji, C.J., 169

Ji, G., 136

Jia, Y., 223–224, 236

Jiang, J., 31–36, 121–131, 217–221

Jiang, X.F., 136

Jin, W.Y., 169–174, 217–221

Jin, Y.F., 223–230, 234

Jirsa, V.K., 158

Johnson, L.R., 320, 328

Jolly, M.R., 95

Jones, R., 334

Joshi, A., 96

Juang, J.-N., 409–410

Jung, P., 233

Junjiro, O., 357–358

Just, W., 169

Kalmar-Nagy, T., 135

Kamath, G.M., 95

Kaminish, K., 233

Kanai, K., 334

Author Index 453

Kanehisa, M., 203

Kang, J.-K., 297

Kao, C.Y., 136

Kapitaniak, T., 31–32, 49

Kappaganthu, K., 105–120

Karabalis, D.I., 331

Karasudhi, P., 324, 332

Karnopp, D.C., 50

Karvinen, T., 285–297, 299–316

Kausel, E., 320, 331–332

Kawakami, H., 32

Kazarinoff, N.D., 159, 163, 165

Kazuyuki, A., 158

Ke, S.Z., 236

Keblinski, P., 101–102

Keer, L.M., 324, 332

Kelley, J.L., 440

Kenji, M., 357–358

Kerschen, G., 275

Keskinen, E.K., 285–297, 299–316

Keskiniva, E., 300, 307

Khadem, S.E., 96

Khadra, A., 203

Khalil, H.K., 119

Kilbas, A.A., 3

Kim, N.H., 135

Kim, Y.B., 121

Kimura, H., 297

Kireitseu, M., 96

Kissenpfenning, J.F., 330

Kleeberger, M., 291

Klein, E.J., 135

Klein, M., 31

Kloeden, P.E., 191

Knospe, C., 122

Kobayashi, K., 31

Kobayashi, O., 32

Kobayashi, S., 320

Kobori, T., 323–324

Kohmuench, J.N., 422

Kong, L., 293

Koratkar, N.A., 96, 101–102

Kordt, M., 384

Kovacs, W.D., 334

Kraincanic, I., 287–289

Krautkr¨amer, H., 341

Krautkr¨amer, J., 341

Krizek, R.O., 333

Krodkiewski, J.M., 122, 263–264

Kuhlemeyer, R.L., 331

Kukla, S., 372

Kumaniecka, A., 291, 297

Kunichika, T., 158

Kuo, C.-H., 417

Kurdi, M.H., 135

Kurths, J., 217

Kusakabe, K., 323–324

Kwon, O.M., 136

Kwon, T., 122

Kypriandis, I.M., 106

Lacovoni, G., 23

Lai, C.H., 217

Lai, Y.C., 32

Laing, C.R., 385

Lainscsek, C., 32

Lakshmikantham, V., 204, 210

Lamb, H., 320, 322, 328

Lampaert, V., 97

Lankarani, H.M., 358

Lardies, J., 276

Launis, S., 291, 300, 313

Lawrence, F.V. Jr., 334

Lee, C.H., 13

Lee, H.C., 297, 410

Lee, K.Y., 375

Lee, S.L., 324, 332

Lee, S.M., 136

Lee, T.H., 333

Lei, P.-F., 61–76, 385

Lei, S., 13

Leiber, T., 233

Lenci, S., 177

Leng, Y.-G., 251–259

Lenz, J., 358

Lesieutre, G.A., 95

Leung, A.Y.T., 169, 264

Levy, V.J., 23

Lewis, F.L., 143–144

Li, B.H., 394

Li, C., 203

Li, G., 177–178, 184

Li, H.K., 397–408

Li, J.L., 223–224

Li, J.R., 236

Li, K.-L., 61–76

Li, K.T., 386

Li, T., 13

Li, W., 143

Li, X., 160–162

Li, Z.G., 204, 234

Liang, G.Y., 224, 234

Liang, Z.Y., 96

Liao, W.H., 95

Liao, X., 203

Liberzon, D., 136, 145–146, 191

454 Author Index

Lin, C.L., 136

Lin, L., 224

Lin, W.W., 13, 204

Lindler, J.E., 95

Liu, A., 96

Liu, B., 203, 251

Liu, G., 342

Liu, H.H., 169–174, 398, 401

Liu, J., 291

Liu, S.D., 19

Liu, S.S., 19

Liu, X.T., 371

Liu, X.Z., 203–204

Liu, Y., 23–29, 95, 384–385

Liu, Y.S., 204, 210

Liu, Y.-X., 371–380

Liu, Z., 19, 23

Ljung, L., 409

Logan, K., 397

Long, Q., 234

Long, S.R., 251, 398, 401

Long, Y.J., 50

Lorenz, E.N., 13, 38

Lou, J.J., 49–60

Love, A.E.H., 285, 287

Lowe, M., 341, 352

L¨u, J., 203, 205

Lu, J.A., 203–205

Lu, K., 385

L¨u, L.-F., 371–380

Lu, Q.S., 158, 170, 178

Lu, X.M., 136

Luco, J.E., 324, 330, 332–333, 335

Lumer, E.L., 170

Luo, A.C.J., 37–47, 77–93, 189–200

Luo, Q., 136

Luo, X.L., 234

Luo, X.Q., 224

Luo, Y., 357–370

Lusebrink, H., 384

Luther, H.A., 140

Luttrell, G.H., 422–423

Lv, C., 403

Lysmer, J., 330–331

Ma, H., 135

Ma, J., 287

Ma, X.J., 400–401

MacCalden, P.B., 333, 335

Maccari, A., 178

Mahajan, A., 409–435

Mahmoodi, S.N., 96

Maistrenko, Y., 32

Majee, P., 224

Mallat, S., 279–280

Mallik, A.K., 50, 56

Mamoon, S.M., 321

Mance, V., 23

Mandel, L., 233

Manley, O., 264, 385

Mann, B.P., 135

Mannell, R., 234

Marion, M., 264, 384–385

Maritan, A., 217

Marotto, F.R., 37

Masoller, C, 169–170

Matsumoto, T., 31

Matthies, H.G., 265

Matthiesen, R.B., 333, 335

Maxwell, A.A., 334

Mccarthy, B., 99

McClintock, P.V.E., 234

McEvilly, T.V., 334

McFadden, P.D., 402

McRobie, A., 385

McRobie, F., 177

Meguid, S.A., 287, 289–290

Mei, D.C., 224, 234, 239

Mei, G.-H., 383–394

Meirovitch, L., 371

Meiss, J.D., 37

Meyer, M., 265

Miah, H., 264

Michaelis, M., 280

Mickens, R.E., 276–277

Miliou, A.N., 106

Miller, D.E., 136

Miller, G.F., 323

Minai, R., 323–324

Minor, G.R., 333

Mohanty, M., 421–435

Mohri, A., 297

Moiseuev, N., 347

Montonen, E., 300

Montonen, J., 291, 299–316

Moon F., 177–178, 184

Moore, R.E., 446

Morse, A.S., 136, 145–146, 191

Mosekilde, E., 37

Moss, F.E., 234, 236

Mroczkowski, T.T., 236

Mu, J., 341

Murota, K., 384

Muszynska, A., 121

Author Index 455

Nagakagi, S., 297

Nagumo, J., 157

Nakagawa, T., 297

Nakano, H., 320

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